Topological Crystalline Phases in a Disordered Inversion-Symmetric Chain

TL;DR

This paper investigates how disorder that preserves inversion symmetry affects the topological phases of a 1D chain, revealing that the inversion topological invariant can fluctuate and is determined by states at inversion centers.

Contribution

It introduces a basis-independent method to analyze topological invariants in disordered inversion-symmetric chains and compares their behavior under strong disorder.

Findings

Inversion topological invariant fluctuates when the spectral gap closes at strong disorder.

Inversion eigenvalues at inversion centers determine the invariant in disordered systems.

The invariant can be altered by occupying states at inversion centers without changing bulk polarization.

Abstract

When translational symmetry is broken by bulk disorder, the topological nature of states in topological crystalline systems may change depending on the type of disorder that is applied. In this work, we characterize the phases of a one-dimensional (1D) chain with inversion and chiral symmetries, where every disorder configuration is inversion-symmetric. By using a basis-independent formulation for the inversion topological invariant, chiral winding number, and bulk polarization, we are able to construct phase diagrams for these quantities when disorder is present. We show that unlike the chiral winding number and bulk polarization, the inversion topological invariant can fluctuate when the bulk spectral gap closes at strong disorder. Using the position-space renormalization group, we are able to compare how the inversion topological invariant, chiral winding number and bulk polarization behave at low energies in the strong disorder limit. We show that with inversion symmetry-preserving disorder, the value of the inversion topological invariant is determined by the inversion eigenvalues of the states at the inversion centers, while quantities such as the chiral winding number and the bulk polarization still have contributions from every state throughout the chain. We also show that it is possible to alter the value of the inversion topological invariant in a clean system by occupying additional states at the inversion centers while keeping the bulk polarization fixed. We discuss the implications of our results for topological crystalline phases in higher-dimensional electronic systems, and in ultra-cold atomic systems.